Wavefront reconstruction in digital off-axis holography via sparse ...

May 15, 2015 / Vol. 40, No. 10 / OPTICS LETTERS

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Wavefront reconstruction in digital off-axis holography via sparse coding of amplitude and absolute phase V. Katkovnik,1,3 I. A. Shevkunov,2 N. V. Petrov,2,* and K. Egiazarian1 1

2

Department of Signal Processing, Technology University of Tampere, Tampere, Finland Department of Photonics and Optical Information Technology, ITMO University, St. Petersburg, Russia 3

e-mail: [email protected] *Coressponding author: [email protected] Received February 4, 2015; revised April 21, 2015; accepted April 29, 2015; posted April 30, 2015 (Doc. ID 233776); published May 15, 2015 This work presents the new method for wavefront reconstruction from a digital hologram recorded in off-axis configuration. The main feature of the proposed algorithm is a good ability for noise filtration due to the original formulation of the problem taking into account the presence of noise in the recorded intensity distribution and the sparse phase and amplitude reconstruction approach with the data-adaptive block-matching 3D technique. Basically, the sparsity assumes that low dimensional models can be used for phase and amplitude approximations. This low dimensionality enables strong suppression of noisy components and accurate revealing of the main features of the signals of interest. The principal point is that dictionaries of these sparse models are not known in advance and reconstructed from given noisy observations in a multiobjective optimization procedure. We show experimental results demonstrating the effectiveness of our approach. © 2015 Optical Society of America OCIS codes: (030.4280) Noise in imaging systems; (070.2025) Discrete optical signal processing; (090.1995) Digital holography; (100.3010) Image reconstruction techniques; (100.3190) Inverse problems; (100.5070) Phase retrieval. http://dx.doi.org/10.1364/OL.40.002417

Digital holography has a huge number of practical applications in optical metrology, industry, medicine, and science, such as shape and deformation measurements down to a fraction of a micrometer or absolute distance measurements on an environmental scale [1]. In digital holography, there are specific problems, such as zeroorder and conjugate images, sampling, and sensitivity to vibration of optical components. Many different techniques have been proposed to solve some of these problems. For instance, the in-line phase shifting digital holography (PSDH) method [2] is free from the presence of zero-order and conjugate images, but usually requires at least three hologram measurements. Contrary to PSDH, off-axis digital holography [3] provides a lower resolution of the reconstructed wavefront due to the necessity for carrier fringes, but allows dynamic applications because of a singe exposure. Another feature of most of the existing digital holography methods is the original formulation of the problem not taking into account the noise occurring at hologram registration. This leads to the fact that on many occasions the reconstructed phase distribution requires additional procedures of noise filtering [4]. The evidence that the wavefront may be a subject of sparse representation has found wide application in digital holography due to the possibility of using compressive sensing (CS) theory and efficient algorithms [5]. This allows wavefront reconstruction from undersampled holograms in the terahertz frequency [6–8], where it is motivated by an existing need to reduce demanding scans performed by a single-pixel detector. In the visible range CS is applied for various microscopy applications [9,10] and imaging with improved resolution [11]. In this Letter, we propose a new iterative technique for off-axis wavefront reconstruction with efficient noise suppression. It is based on the powerful nonlocal Block-Matching and 3D (BM3D) imaging technique [12] using patch-wise grouping, spectral analysis of groups, 0146-9592/15/102417-04$15.00/0

thresholding of group-wise spectra, and collaborative inverse data processing. It is recognized that the BM3D technique demonstrates the state-of-the-art performance for various applications. Sparse representations of images imply that they are well approximated by linear combinations of a small number of functions taken from known sets (dictionaries). On many occasions, this is a consequence of the self-similarity of images: it is very likely to find in them many similar patches in different locations and at different scales. By analogy with the conventional imaging the sparse representation is possible and efficient for complex-valued wavefronts, as shown in [13] at the example of the BM3D technique. Consider v ∈ Cn object wavefront, which is given in the form v
Bv expjφv  on a grid with n pixels. Let us assume that amplitude Bv and absolute phase φv;abs (obtained after unwrapping of principal phase φv , φv;abs
W −1 φv , where W −1 stays for unwrapping operator) admit a sparse representation, or sparse coding. So we can write Bv
Ψa;v θa;v ;

φv;abs
Ψφ;v θφ;v ;

(1)

θa;v
Φa;v Bv ;

θφ;v
Φφ;v φv;abs :

(2)

Here matrices Ψc;v ∈ Rn×m (where c means a or φ) are termed synthesis operators (or dictionary) and θc;v ∈ Rm are vectors, namely, the amplitude and phase (absolute phase) spectra of the object v for c
a and c
φ, respectively. In Eq. (1), Bv and φv;abs are synthesized as linear combinations of the columns of Ψc;v weighted by the elements of θc;v . The synthesis-based representations have a dual point of view in which, for given amplitude Bv ∈ Rn and absolute phase φv;abs ∈ Rn , we compute their spectra θc;v ∈ Rm by applying the so-called analysis operator © 2015 Optical Society of America

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OPTICS LETTERS / Vol. 40, No. 10 / May 15, 2015

(or dictionary) Φc;v ∈ Rm×n to Bv and φv;abs , as shown in Eq. (2). Following the rationale of the sparsity, we herein assume that θc;v are sparse, i.e., most elements thereof are zero. For sparse modeling of phase and amplitude we use the BM3D technique. It is shown in [13–15] that BM3D grouping of patches, analysis, synthesis, and thresholding can be combined in a single procedure named BM3D. The input-output representation of the BM3D algorithm for phase and amplitude can be given in the following compact form: φˆ v;abs
BM3Dφv;abs ; τφ ;

(3)

Bˆ v
BM3DBv ; τa ; where the inputs φv;abs and Bv are the phase and amplitude images of the object wavefront and the outputs ˆ v;abs and Bˆ v images are the sparse representations φ (approximations) of the corresponding inputs. Here τφ and τa are the threshold parameters. Larger values of these parameters mean stronger filtering (smoothing) properties of the filters. In order to quantify the level of sparsity of θc;v , we use the convex l1 -norm ‖θc;v ‖1 defined as the sum P of absolute values of the elements of θc;v , thus, ‖θc;v ‖1
ns
1 jθc;v js [16]. The corresponding thresholding operation is applied to the elements of the spectral variables and defined by the formula θˆ c;v
Thτ θc;v 
signθc;v  · maxjθc;v j − τc ; 0;

(4)

where τc is the thresholding parameter as denoted in Eq. (3). This thresholding preserves all variables such that jθc;v j ≥ τ and zeros all others, but the absolute values of the preserved variables are replaced by smaller values jθc;v j − τ. Naturally, the success of the sparse imaging depends on how reach and redundant are the dictionaries Φc;v and Ψc;v used for the analysis and synthesis of absolute phase and amplitude. In BM3D the dictionary design is imbedded in this algorithm. Both the analysis and synthesis dictionaries are obtained from given noisy observations. These dictionaries are nonlocal, data adaptive, and as shown in experiments, very efficient for sparse approximations. This data-adaptive dictionary design is one of the key points of success and advantage of BM3D algorithms as compared with standard bases, such as Fourier or Fresnel conventionally used in many applications (e.g., [17,18]). On many occasions the BM3D dictionaries also demonstrate advantage over the learned dictionary techniques based on the dictionary obtained from external databases with test images typical for specific applications [16]. The considered optical setup is clear from Fig. 1. Here the bitelecentric system projects the object on the sensor plane. In this way we arrive to the so-called image-plane holography, where the object image is reconstructed at the sensor plane. In this Letter we are concentrated on the image-plane holography because it is more straightforward for demonstration of advantages of the sparse modeling of the wavefront. Note the digital Fresnel holography contrary to the considered setup requires

Fig. 1. Experimental holography.

setup

for

off-axis

image-plane

backward/forward wavefront propagation as additional processing steps to reconstruct the object. As is well known, the off-axis hologram is the interference pattern I of object uo
Bo expjφo  and reference ur
Ar expjφr  wavefronts, measured in the registration (sensor) plane as I
jBo expjφo   Ar expjφr j2 :

(5)

In case of plane reference wavefront the reference phase φr can be estimated as φr
2πx sin αx  y sin αy ∕λ;

(6)

where αx and αy are angles of the reference beam plane with respect to the optical axis z, x; y are coordinates in the sensor plane, and λ is the wavelength. Thus, the reference beam at the sensor plane can be synthesized as a 2D sin α harmonic function of the frequencies sinλ αx , λ y on x and y, respectively. Following [19] we introduce auxiliary variables: U
jBo 2  jAr j2

and

Z
uo Ar :

(7)

The key point of this replacement is that the intensity I, originally quadratic with respect to the amplitudes Bo and Ar , becomes linear function on the new variables U and Z. Then I can be rewritten as I
U  V  Z  Z  V ;

(8)

where V
expjφr . If U and Z are given the amplitude Ar is computed according to the formula [19] p1∕2  U  U 2 − 4jZj2 ; Ar
2

(9)

where the plus is used for juo j < Ar and the minus for juo j > Ar , and then uo
Z∕Ar :

(10)

The approach developed in [19] is based on the leastsquare solution of Eq. (8) assuming that the variables Bo φo , Ar are invariant in a neighborhood of each point of observation. A larger size of the neighborhood results in a lower resolution of imaging and better noise suppression. A proper selection of the neighborhood is a serious problem of this technique as it is discussed in [19] in the

May 15, 2015 / Vol. 40, No. 10 / OPTICS LETTERS

context of image resolution but without analysis of the noise effects. Our approach is different in two principal aspects. First, we start from the variational setting of the problem targeted on the optimal noise suppression. Second, the hypothesis of the point-wise invariant variables used in [19] is replaced by the hypothesis that these variables are sparse. This assumption is much more flexible and not so restrictive as the one used in [19], and the default allows to get more accurate wavefront reconstructions. It is assumed in our approach that the observations of I are noisy and defined by the equation Y
I  σε;

(11)

where ε is the independent and identically distributed standard zero-mean Gaussian noise and σ is the standard deviation of the noise. Then following the maximum likelihood paradigm, the sparse hologram reconstruction can be formalized as the following optimization problem: minLY ; I  ‖θa;Z ‖1  α1 ‖θφ;Z ‖1  α2 ‖θU ‖1 ; U;Z

(12)

where LY ; I is the fidelity term defined as the minus log-likelihood function of the random Y , and ‖θa;Z ‖1 , ‖θφ;Z ‖1 and ‖θU ‖1 are the sparsity criteria, respectively, for the amplitude of Z, the phase of Z, and the real valued U; α1 , α2 > 0 are weighting parameters. Thus, we try to maximize the sparsity by minimizing a number of nonzero elements of the spectra of the corresponding variables. For the complex-valued Z, the sparsity is defined separately for the phase and amplitude. While there are techniques for dealing with the problems in the form Eq. (12) we prefer a much more manageable formulation of optimization where special splitting variables separate denoising and sparsification operations. Therefore, we adopt an optimization approach [12]. The main intention of the approach is simultaneous minimization of the minus log-likelihood function of observations (maximum likelihood approach) and the l1 -norms of amplitude and phase spectra supporting the sparsity of the estimates of these variables. Following the development in [15] we define the estimates of Z and U by minimizing the criterion 1 X U − W U 2 J U; Z
LY ; I  2γ 0 m 1X  jZ − W Z j2 ; (13) γ1 m P where LY ; I
2σ1 2 m Y − U  V  Z  Z  V 2 is minus log-likelihood defining the fidelity term for Gaussian observations, W U and W Z are the splitting variables having the sense of the sparse approximations of U P and Z, respectively. The summation is calculated m overall pixels of the grid and for the sake of notation simplicity the arguments are omitted in the phase and amplitudes of the object and reference wavefronts. These sparse approximations are built from U and Z using the BM3D procedures Eq. (3). Minimization of Eq. (13) gives the result in the form

ˆ Z ˆ
arg U;

min JU; Z:

U>0;Z⊂C n

2419

(14)

Note that in this minimization the solution depends on the variables W U and W Z only: ˆ Uˆ
UW U ; W Z ;

ˆ U ; W Z : Zˆ
ZW

(15)

Combining the representation of the solution of Eq. (14) in the form Eq. (15) and using BM3D procedures Eq. (3), the sparse phase and amplitude reconstruction (SPAR) algorithm for the off-axis holography can be given in the form as presented in Fig. 2. ˆ 0U and W ˆ 0Z , which are The algorithm is initialized by W the estimates of U and Z obtained as output of the algorithm presented in [19] derived, as it is mentioned above, for the windowed point-wise least square approximation of the noisy observations. In Step 2, W −1 stays for the unwrapping operation by the PUMA algorithm [20]. The sparsity assumption from the very beginning can be imposed on the variables of the interest: object wavefront, Bo , φo , and amplitude Ar of the reference wavefront. However, we found that the approach presented above, with the sparsity of variables U and Z, results in a more accurate algorithm where the sparsity assumption on Bo and Ar (BM3D filtering) are applied only for the final estimates, obtained when the iterations are terminated, after Step 4 of the algorithm. The developed algorithm was applied to the holographic data obtained for a fly’s wing used as a specimen, image size 768 × 768. Selected amplitude-phase object demonstrates proof of the principle of the SPAR algirithm both for amplitude and for phase object reconstructions. The following parameters of the algorithm are used in our experiments: the laser wavelength λ
634.9 nm and the sensor-pixel pitch Δ
2.8 μm. The phase of the reference wavefront is defined by Eq. (6), where αy
0

Fig. 2.

SPAR algorithm for the off-axis holography.

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digital off-axis holography is able to produce high-quality imaging for quite noisy data. Sparse modeling of amplitude and absolute phase is one of the key elements of this advanced performance. This work is supported by the Academy of Finland, projects 138207, 2011-2014, and the Russian Ministry of Education and Science project within the state mission for institutions of higher education (agreement 2014/ 190). The authors thank Arkadiy Drozdov for help in preparing illustrations. Fig. 3. Obtained reconstructions of the object (a) amplitude, (b) wrapped, and (c) absolute phase, and vertical cross sections of absolute phases (top), and amplitudes (bottom) for SPAR algorithm (red) in comparison with the initial estimates obtained by [19] (black).

and αx ≈ 3.25 deg . At Step 1 the initialization algorithm from [19] is used with the Gaussian weight function, σ 1
1 and the window size 7 × 7. We have kept BM3D parameters for synthesis and analysis fixed [14,15]. The size of the image patches is 8 × 8 and the group size is limited by the number 8. The step size between the neighboring patches in grouping is equal to 3. The parameters γ 0 and γ 1 appearing in J U; Z and the thresholds τφ , τU , and τjW Z j used in Steps 3 and 7 of the algorithm are set to the following heuristic basic values: γ 1
σ 2 · 0.2∕1  0.2 · t, γ 0
5γ 1 , and τφ
1.4∕1  0.5 · t; τU
τjW Z j
1∕1  t, where t is an iteration number. Note that smaller values of γ 0 , γ 1 and larger values of the threshold result in stronger smoothing properties of the algorithms. Figure 3 shows the SPAR reconstructions (second line) in comparison with the reconstructions given by the algorithm from [19] (first line). The following images are shown: object amplitude (a), object wrapped phase (b), and unwrapped (absolute) phase (c). The SPAR reconstructions look like sharp, clear images essentially denoised in comparison with the quite noisy reconstructions obtained by [19]. In order to make the advantage of the SPAR reconstruction obvious in Fig. 3(d) we show the vertical cross sections (column number 400) of the images in Figs. 3(a) and 3(c). It shows how accurately the SPAR algorithm eliminates the noise and preserves important phase-image details. We may conclude, based on the demonstrated results, that the novel variational algorithm developed for the

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